6.4. EXERCISES 127

Show that for W (x) =W (y1, · · · ,yn)(x) to save space,

W ′ (x) = det

y1 (x) · · · yn (x)y′1 (x) · · · y′n (x)

......

y(n)1 (x) · · · y(n)n (x)

 .

Now use the differential equation, Ly = 0 which is satisfied by each of these func-tions, yi and properties of determinants presented above to verify that

W ′+an−1 (x)W = 0.

Give an explicit solution of this linear differential equation, Abel’s formula, and useyour answer to verify that the Wronskian of these solutions to the equation, Ly = 0either vanishes identically on (a,b) or never. Hint: To solve the differential equation,let A′ (x) = an−1 (x) and multiply both sides of the differential equation by eA(x) andthen argue the left side is the derivative of something.

47. Find the following determinants and the inverses of the given matrices. You mightuse MATLAB to do this with no trouble.

(a) det

 2 2+2i 3−3i2−2i 5 1−7i3+3i 1+7i 16

 (b) det

 10 2+6i 8−6i2−6i 9 1−7i8+6i 1+7i 17

